Formulas of Viete

Posted January 17th, 2008 by Structure

$a_0x^n+a_1x^{n-1}+a_2x^{n-2}+...+a_{n-1}+a_n=0$
Let $x_1, x_2,...x_n$ the roots of this equation.
Then

$\sum_{i=1}^n<br /> =x_1+x_2+...+x_n=(-1)^1\frac{a_1}{a_0}$

$\sum_{1\leq i_1<i_2\leq n}x_{i_1}x_{i_2}=x_1x_2+x_1x_3+...+x_{n-1}x_n=(-1)^2\frac{a_2}{a_0}$

$\sum_{1\leq i_1<i_2<..i_3\leq n}x_{i_1}x_{i_2}x_{i_3}=x_1x_2x_3+x_1x_2x_4+...+x_{n-2}x_{n-1}x_n=(-1)^3\frac{a_3}{a_0}$

$\sum_{1\leq i_1<i_2<..i_3<....<i_n\leq n}x_{i_1}x_{i_2}x_{i_3}....x_{i_n}=x_1x_2x_3....x_{n-1}x_n=(-1)^n\frac{a_n}{a_0}$

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